1. Field of the Invention
The present invention relates to a stable equilibrium point calculation apparatus which calculates a stable equilibrium point of a system, and more particular to a BCU classifier (Boundary of stability-region-based Controlling Unstable equilibrium point classifier) system which can classify contingencies more exactly and quickly than a conventional BCU classifier system.
2. Description of the Related Art
A stable equilibrium point of a nonlinear dynamic system is obtained by solving a multidimensional nonlinear equation. Thus, usually, a Newton method, which makes use of a well-known Jacobian matrix, is well known.
Since the Newton method performs a convergence calculation by the Jacobian matrix, there are some cases in which convergence is not attained. In order to solve this problem of non-convergence, various methods have been proposed (reference documents 1 and 2). These proposed methods include a pseudo-transient simulation method (reference documents 3 and 4).    Reference document 1: S.-C. Fang, R. C. Melville, L. Trajkovic, and L. T. Watson, “Artificial parameter homotopy methods for the dc operating point problem,” IEEE Trans. Comput.-Aided Des. Integr. Circuits syst., vol. 12, No. 6, pp. 861-877, June 1993.    Reference document 2: T. L. Quarles, SPICE 3C.1 User's Guide. Berkeley: Univ. California, EECS Industrial Liaison Program, April 1989.    Reference document 3: T. S. Coffey, C. T. Kelley, and D. E. Keyes, “Pseudo transient continuation and differential algebraic equations,” SIAM J. Sci. Comput., vol. 25, No. 2, pp. 553-569.    Reference document 4: C. T. Kelley and D. E. Keyes, “Convergence analysis of pseudo-transient continuation,” SIAM J. Numer. Anal., vol. 35, No. 2, pp. 508-523, 1998.
To find a stable equilibrium point (SEP) of a nonlinear dynamic system is important in order to confirm the stable operation of the system. In order to find the stable equilibrium point, a nonlinear simultaneous equation is solved. As regards a nonlinear simultaneous equation, it is difficult to analytically obtain a solution, so that a Newton method is generally used. In the Newton method, a Jacobian matrix is utilized and linear simultaneous equations are repeatedly solved, and thereby a solution is found. Since this method is simple, the method is used in a wide field.
In a transient stability screening program (BCU method), too, a stable equilibrium point is calculated by a Newton method since it is important to analyze the property of the stable equilibrium point. In a case where a stable equilibrium point cannot be found, there is no choice but to determine that the system is very unstable, and a detailed-time-domain simulation which involves a great amount of computation has to be performed. Thus, a great deal of labor is needed in order to determine contingencies which are classified by the stable equilibrium point convergence problem of the BCU classification method.
Although the Newton method is an easy-to-handle, highly practical method, numerical divergence occurs if initial values are not proper (i.e. if initial points are not present in a region where convergence is possible). No index is given for determining whether convergence is not attained due to numerical divergence, or there is, actually, no stable equilibrium point. If there is no stable equilibrium point, the danger of system collapse is considerably high and a proper measure is needed. In a case where it is unclear whether a stable equilibrium point is actually present or not, an appropriate calculation procedure for evaluating a stability of the power system can not be executed.
A problem in a case where a stable equilibrium point cannot be found by the Newton method may become a fatal defect, in particular, in a case where transient stability evaluation needs to be carried out on line, as in the case of BCU method.
A nonlinear simultaneous equation is a static calculation equation, and inherently it does not have dynamic characteristics. However, there is known a method (Pseudo-Transient Method) in which virtual dynamic characteristics are assumed in the nonlinear simultaneous equation, and dynamic simulation (an implicit integration method with a variable integral time step is used) is performed, thereby obtaining approximation of a stable equilibrium point. Since the pseudo-transient method is based on a simulation method, it is hardly possible to completely attain convergence to a stable equilibrium point. Thus, if the integral time step increases to a certain degree, it is assumed that the vicinity of a stable equilibrium point is reached, and a convergence point is found by the Newton method (FIG. 1). In a case where there is no stable equilibrium point, some features, for instance, such a feature that the integral time step does not increase, would appear. Accordingly, distinction becomes clear between a problem relating to numerical analysis and a case in which there is, actually, no stable equilibrium point.
In many cases, the pseudo-transient simulation method effectively exhibits desired effects. However, while the pseudo-transient simulation method was being applied to the stable equilibrium point calculation of a classifier II of an improved BCU classification method (reference document 5) of a TEPCO-BCU method, it was made clear that if the attenuation of the system is not large, many integral calculations are required in order to calculate a stable equilibrium point, and that if vibration properties are high, a stable equilibrium point cannot properly be found.    Reference document 5: U.S. Pat. No. 6,868,311.